Mixed Tate motives and cyclotomic multiple zeta values of level $2^n$ or $3^n$

Let $N$ be a power of $2$ or $3$, and ${\mu }_N$ the set of $N$-th roots of unity. We show that the ring of motivic periods of Mixed Tate motives over $\mathbb{Z}[{\mu }_N,\frac{1}{N}]$ is spanned by the motivic cyclotomic multiple zeta values of level $N$. This implies that the action of the motivic Galois group of mixed Tate motives over $\mathbb{Z}[{\mu }_N,\frac{1}{N}]$ on the motivic fundamental group of ${\mathbb{G}}_m-{\mu }_N$ is faithful. This is a generalization of the known results for $N\in \{1,2,3,4,8\}$ by Deligne and Brown. We also discuss cyclotomic multiple zeta values of weight $2$ of other levels.